This is the title of a talk given at the Seminář z algebry at Masaryk university on March 16, 2017. The talk presents the corresponding two papers.
Universal classes are a general model-theoretic framework introduced in the seventies by Saharon Shelah to study certain classes of modules. It encompasses several natural algebraic objects, such as vector spaces and locally finite groups.
I will present generalizations to universal classes of several concepts and results of linear algebra. For example, a universal class which has a single model of a "high-enough" infinite size has a single model in every high-enough size. Moreover, such classes admit an independence notion generalizing linear independence in vector spaces. I will also discuss a more general framework (also due to Shelah), abstract elementary classes, and conjectured extensions of these results there.
John T. Baldwin, Categoricity, University Lecture Series, vol. 50, American Mathematical Society, 2009.
Saharon Shelah, Classification Theory for Abstract Elementary Classes, Studies in Logic: Mathematical Logic and foundations, vol. 20, College Publications, 2009.
Sebastien Vasey, The lazy model theoretician's guide to Shelah's eventual categoricity conjecture in universal classes, An expository note: pdf arXiv.
Sebastien Vasey, Shelah's eventual categoricity conjecture in universal classes. Part I, Accepted, Annals of Pure and Applied Logic. Preprint: pdf arXiv.
Sebastien Vasey, Shelah's eventual categoricity conjecture in universal classes. Part II, Accepted, Selecta Mathematica. Preprint: pdf arXiv.
Sebastien Vasey, Shelah's eventual categoricity conjecture in tame AECs with primes, Accepted, Mathematical Logic Quarterly. Preprint: pdf arXiv.
Will Boney and Sebastien Vasey, A survey on tame abstract elementary classes, Accepted, Beyond First Order Model Theory (José Iovino ed.), CRC Press. Preprint: pdf arXiv.